The Top 2 Math Stories of 2006

The math discovery of the century (already!), math that's in your ear.

If, in the year 2100, DISCOVER runs a feature on the top advances in science in the 21st century, the proof of the Poincaré conjecture is still likely to be the number-one story in mathematics.

Proposed in 1904 by the French mathematician Henri Poincaré, the conjecture describes a way to determine the structure of space. All attempts at a solution failed until 2002 and 2003, when an established but reclusive Russian mathematician named Grigori Perelman posted three papers on the Internet, claiming that they provided the key steps for proving a general conjecture (the geometrization conjecture) formulated by the American mathematician William Thurston in the 1970s. Although Perelman did not mention it by name, the Poincaré Conjecture is a direct consequence of the Geometrization Conjecture. The 58-page proof Perelman posted online was so original, and so compressed, that it took the world's experts almost four years of work to produce a deconstructed 500-page version that convinced them his work was correct.

At the 2006 International Congress of Mathematicians, held in Madrid in August, Perelman was named a winner of the Fields Medal, the mathematical world's equivalent of the Nobel Prize. Yet the publicity-averse Perelman not only declined to attend the conference, he also refused to accept the medal.

To appreciate the significance of Poincaré's conjecture about the structure of space, consider the analogy of a two-dimensional ant living on (actually, in) a surface—a two-dimensional universe. From the outside, we see the creature's universe as a surface. It could be the surface of a sphere, a torus (a doughnut-shaped surface), a two-holed torus, and so on. But would the ant living in the surface have any way of knowing its shape? For each of those possible shapes, after all, the ant has exactly the same freedom of movement, forward and backward or left and right, for as far as it likes.

An analogous question arises for us in our three-dimensional universe. We can travel forward and backward, left and right, up and down, as far as we choose. But as with the ant, this freedom of movement does not tell us the shape of the space we inhabit. Since we can't step outside our universe and look down on it, is there any way we could determine its shape from the inside? Poincaré proposed a method whereby we might be able to answer this question.

Imagine we set off on a long spaceship journey. (This is not how Poincaré formulated the conjecture; he used the formal mathematical language of topology and geometry.) As we travel, we unreel a long string behind us. After journeying a long while, we decide to head home—and not necessarily along the same path. When we return to our starting point, we make a slipknot in our string and begin pulling the string through. One of two things can then happen: Either the noose eventually pulls down to a point, or it reaches a stage where it cannot shrink any further, no matter how much we pull.

Poincaré's conjecture is that if we can always shrink the loop to a point, then the space we live in is the 3-D analogue of the surface of a sphere; but if we can find a start-end point and a journey where the loop cannot be shrunk to a point, then our universe must have one or more "holes," much like a torus. (Essentially, what happens is that the string snags by going around such a hole.)

For our two-dimensional ant, it is easy to prove that such a procedure will enable the creature to determine from the inside whether its universe is like the surface of a sphere or whether there are holes in it. But until Perelman came along, nobody had proved that the same technique would work in our three-dimensional universe and allow us to tell from the inside what kind of universe we live in. We now know that we can. In fact, the more general geometrization conjecture that Perelman's argument established tells us we can, in principle, determine a great deal more about the shape of the universe.

The interest to scientists is clear, but will the new result have any impact on everyday life? The answer is almost certainly yes, although that impact will most likely come from the methods Perelman developed to solve the problem rather than from the result itself. Many of the fundamental equations physicists use to understand the universe have so-called singularities, places where they produce infinite (and hence useless) answers. To prove the Poincaré conjecture, Perelman developed powerful methods to handle potential singularities in the equations of space. Those techniques are certain to lead to major developments in physics. And history tells us that major advances in physics will, in due course, change life on Earth forever.

Keith Devlin

74. Cochlea's Spiral Plays Surprising Role in Hearing

Deep inside your ear, the pea-size, spiral-shaped cochlea helps translate reverberations from the outside world into neurological signals that we perceive as sound. The cochlea's coil has traditionally been regarded as little more than the body's way of packing a lot of membrane into a small space—a mechanical adaptation that did not affect hearing. Not any more.

Last March a team of engineers found a function for the cochlea's shape. Using a mathematical model, they determined that the tight coil at the cochlea's center steers low-frequency waves into its tightest turns, helping us hear deep vibrations. Previous models had treated sound waves as if they traveled in a straight line, an assumption that failed to take into account how the cochlea's shape affects the waves' path. "It's the curvature that's critical," says biophysicist Richard Chadwick at the National Institutes of Health, a collaborator on the project. "The more the curvature changes, the more focused the energy gets. It's behaving something like a whispering gallery, but even better."